8. Fibonacci and the Golden Section

Suppose you were employed by a publishing firm as a graphic artist, and you were required to present a diagram for a children’s encyclopaedia under the heading ‘Rectangle.” Which of the following designs would you have chosen to illustrate the word?

I guess most people would choose C.’ Why is this?’ Diagram A is too near that of a square.’ D is unnecessarily elongated. B and C are similar, but the eye approves of C because it is a ‘horizontal’ rectangle, rather than a ‘vertical’ rectangle. What is it about C that causes us to choose the design?’ There is something about C that seems to speak of a ‘right proportion’. It is pleasing to the eye. My son-in-law Andrew White is a nationally acclaimed figurative and landscape artist. He obtained his honours degree in fine art at the Slade in London. He tells me that part of the teaching programme in College is focused on the subject of ‘right proportion’, and how it affects paintings. He would more often choose C proportions for a canvas than A or D, and he knows why. Let me explain.

There is something about the working of the human eye that finds satisfaction in the rectangular shape of C. It is known as the Golden Section, and has been employed in both Art and Architecture throughout the ages. The Parthenon in Athens, for example, was built on the basis of the Golden Section. But it is one thing to know and to use the Golden Section, but quite another to understand why it happens. It is a purely subjective anatomical phenomenon, based on the working of the human eye. Our vision is most acute when the rays of light pass through the lens and fall into a small oval shape in the retina which is along the optical axis of the eyeball. This oval shape is called the Fovea Centralis and is less than a millimetre across. The oval is horizontal, and has proportions of 1.6 to 1. In other words, when the image of the rectangle C falls into the Fovea, it is accommodated easily and restfully, giving a pleasing sensation.’

Credit for discovering the truths connected with the Golden Section come from a man by the name ofFibonacci. This Italian mathematician, whose real name was Leonardo of Pisa, 1170 to 1230 A.D. travelled about the Mediterranean coastlands collecting information about mathematics. In 1202 he returned to Pisa and published a book entitled LIBER ABACI, in which he established the introduction of the Arabic notation inEurope; and provided a foundation for future development in arithmetic and algebra. Some years later, in 1220, he published another book called PRACTICA GEOMETRIA, a treatise on Geometry. Out of these two books comes the study of numbers which have ever since been attributed to Leonardo, known as the FIBONACCI SEQUENCE. (The name derives from Filius Bonacci, meaning “son of Bonacci”, Leonardo’s father.) However, the mathematician knew nothing of the fovea centralis, which had to wait until the 20th century.

Before examining the work of Fibonacci, we must ask some important questions. Is it a coincidence that ‘right proportion’ is based upon the structure of the human eye? What if the Fovea was shaped like the rectangle D? Would art and architecture have been based on this? Is ‘right proportion’ merely an expression of the Foveal shape? Or is there something about ‘right proportion’ that is established in nature regardless of the shape of the Fovea? To answer these questions, we must now turn to the Fibonacci sequence.

Fibonacci constructed a sequence of numbers in the following way. Starting with 1, he proceeded to add the previous number to find the next in the series, which in this case was 0. The next number in the sequence was therefore also 1. But the next number would be 1 + 1 = 2, and the following number would be 1 + 2 = 3, and so on. This produced the following series.

1,” 1,” 2,” 3,” 5,” 8,” 13,” 21,” 34,” 55,” 89,” 144,” 233′ . . . . .

He then calculated the ratio of successive numbers and found the following information, shown in tabular format.

1/1

2/1

3/2

5/3

8/5

13/8

21/13

34/21

55/34

89/55

144/89

233/144

1

2

1.5

1.667

1.6

1.625

1.615

1.619

1.619

1.618

1.618

1.618

All ratios beyond 55/34 showed the same answer, namely 1.618. This number is commonly known as the Golden Number, and the ratio of 1.618 to 1 is the exact size of the Golden Section, producing ‘right proportion.”It is beginning to look as though ‘right proportion’ is established in its own right, rather than being the necessary consequence of the shape of our Fovea.

Fibonacci found something else about his sequence. Starting with any two numbers of choice, he showed that after a few initial ratios, the value of 1.618 always appeared. Let us try that with the numbers’ 4 and 9. First of all we must construct a series.

‘Now wait a minute,’ I can hear someone say, ‘just because this Italian mathematician played around with a number sequence doesn’t have any bearing on the problem you placed before us. Okay, he turned up this number 1.618, but so what?’ Why should that establish ‘right proportion’?” I accept that comment, but would suggest waiting to see further developments, based on his sequence.

The numbers 2,’ 3,’ 5,’ 8,’ 13,’ and so on occur in nature in profusion. Take for example fir cones. By examining the way in which the little knobs are arranged, it becomes apparent that they are laid out in two opposing spirals, and by collecting cones from many different trees, one finds cones with a 3-to-5 arrangement, others with 5-to-8, and yet others with 8-to-13.’ Pineapples display the 8-to-13 arrangement.’ But on looking further one finds that the sharp spines on a gorse bush are disposed in whorls where 21 spines occur in 8 revolutions before the sequence is repeated. Then have a look at the centre of a sunflower, with its amazing array of florets, and you will find opposing spirals of 55-to-89. In fact this subject is so important as to have a whole chapter in most biology books under the heading of phylotaxy.’ I have only mentioned a very few examples here, but when our children were young we spent time gathering specimens from many plants and trees, and found it exciting to see that in nature everything seemed to be arranged in terms of the Fibonacci sequence.

It is this discovery that puts the Fibonacci sequence into the picture, not just as a mathematical oddity, but as a basis upon which all plant life is arranged. The artist who uses ‘right proportion’ is therefore marrying in with nature in this respect, because as we have seen, the ratio of 1.618 to 1 is the eventual result of the series.

Keeping this in mind, let us now turn to something else. We have approached the subject thus far in terms of what the eye sees. How about taking a journey into the world of sound and hearing? The human ear is an instrument of great complexity, amazingly designed, and scientists are still bewildered about its ability to transmit sound waves to the brain to give us such clarity of hearing. The inner ear contains a spiral, known as the cochlea, in which there is a fluid and a profusion of tiny hairs that seem to be the sound receptors, and also the means by which we can detect even the very slightest change in pitch, which enables a guitarist, for example, to tune strings so exactly.

Because of our ability to hear, and to distinguish between frequencies of sound, we have been able to create music, and this is based on what is called the ‘diatonic scale’.’ The notes on a piano keyboard are arranged in ‘octaves’ of 8 white notes, and a ‘chromatic scale’ of 13 notes which includes the 5 black notes. Already we have mentioned three of the Fibonacci numbers.’ Now think about the ‘tonic chord’, consisting of Doh, Me, Soh, Doh.’These notes are numbered 1,’ 3,’ 5,’ and 8.’ Or if we think about it in terms of the Chromatic Scale, Doh, Me, Soh, and Doh are numbered’ 1,’ 5,’ 8,’ and 13.’ There is something sweet, harmonious, pleasing to the ear when this chord is sounded.’ And it is the chord which represents the beginning and end of most pieces of musical composition.

Now, I have no idea how this is so, but there must be something within the structure of the ear corresponding to the Fovea in the eye, which grants us this pleasing sensation based on the Fibonacci sequence. The arrangement of all those thousands of tiny hairs in the cochlea is designed so we may appreciate the harmony derived from the Fibonacci sequence. And this throws up the questions posed before. If the cochlea was designed in some other way, would music sound different? Would we have a pleasant sensation from some other arrangement of frequencies?’ Would the diatonic scale give way to some other keyboard arrangement?’But we are beginning to see that there is a strange agreement between what the eye sees, and what the ear hears, and the way in which all nature is constructed. There is a basic symphony of design found in profusion throughout the world of living things. The Fovea and the Cochlea had to be created in this way.

I mentioned that fir cones and pineapples possess opposing spirals. These are called ‘equiangular spirals’, or‘logarithmic spirals’ by mathematicians, and they are also a part of the Fibonacci empire!’ I will not go into how the sequence throws up the mathematics of the spiral, because it would be too complicated for an article of this sort, but I trust you will be able to take my word for it. These spirals are also found in profusion in nature. For example, the way in which a head of hair is formed, most easily observed on babies as their hair begins to grow. The way in which elephant’s tusks are formed, the horns of sheep, the claws of birds, our own finger nails, and so on.’ And then there is the amazing world of sea shells and snails. But even in the inanimate world of spiral galaxies, one finds this mathematical form. I am told that it is also found in the DNA spirals, and therefore from the very largest to the very smallest, the logarithmic spiral is everywhere.

Let me give you one further example to conclude this section. In the world of atomic and nuclear physics, one learns in science lessons that matter is composed of protons and neutrons. ”Atoms contain varying numbers of these two basis building blocks. The following table shows just a few of the chemical elements that comprise the periodic table.

Element

Protons(P)

Neutrons (N)

RatioN/P

Helium

2

2

1

Carbon

6

6

1

Aluminium

13

14

1.076

Iron

26

30

1.154

Silver

47

60

1.277

Gold

79

118

1.494

Uranium

92

146

1.587

Last element in the Transuranic group

118

191

1.618

The last element has not yet been formed by scientists, but can be identified graphically from a plot of Protons against Neutrons, and extrapolating it to the value of 118, which would be the last in the Transuranic sequence. Never mind if this is too ‘scientific’ for you to digest ‘ just take my word for it. But the important thing is the final ratio ‘ it is exactly the ‘golden number’ of 1.618, and so it is found throughout nature, wherever one may look.

I would suggest that the information in this article is probably the most profound example of what is currently known as Intelligent Design. Regardless of what the ‘raging evolutionists’ are proclaiming these days, they can never get free of this profusion of examples of Fibonacci in nature, and will one day have to stand before the Maker, and argue with Him as to why they refused to accept the evidences. Paul told the Roman Christians that‘the invisible things of God from the Creation of the World are clearly seen, being understood by the things that are made, even His eternal power and Godhead, so that they are without excuse’ ‘who refuse to accept the evidence. (Rom.1:20)

The human eye and the human ear were created by God to be able to appreciate the harmonies of the Golden Section, but also to appreciate God’s own great power and creative wonders.’ We are living in wonderful ‘houses’, made by a wonderful God, who deserves our profound worship.

About Arthur Eedle

Arthur was born in 1931, and became a Christian in 1948. At London University he gained a 2nd honours degree in Physics. He went on to get a Teaching Diploma, and throughout his career life taught physics in England, Kenya, and Hong Kong. Coupled with his love of science, he was a keen student of Greek and Hebrew, and gave many lectures on Biblical subjects. Read more